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Spam yourself with math PhD qualifying-exam questions.

UC Irvine Fall 2025, 1 Complex analysis, Schwarz–Pick lemma, upper half-plane, holomorphic self-maps

Let $𝑓 : 𝐻 \to 𝐻$ be a holomorphic map from the upper half-plane to itself. Show that for all $𝑧 \in 𝐻$ ,

| 𝑓^{'} (𝑧) | \leq Im \frac{𝑓 (𝑧)}{Im} 𝑧 .

UC Irvine Fall 2025, 2 Complex analysis, entire functions, dense image, Casorati–Weierstrass

Let $𝑓$ be a non-constant entire function. Prove that the image $𝑓 (ℂ)$ is dense in $ℂ$ . Do not use Picard's theorem.

UC Irvine Fall 2025, 3 Complex analysis, Jensen's formula, zero counting, entire functions

Let $𝑓 (𝑧)$ be an entire function with $𝑓 (0) = 1$ . For $𝑟 > 0$ , let $𝑛 (𝑟)$ denote the number of zeros of $𝑓$ in the closed disk $| 𝑧 | \leq 𝑟$ . Let

𝑀 (𝑟) = max_{| 𝑧 | = 𝑟} | 𝑓 (𝑧) | .

Prove that for any $𝑟 > 0$ ,

\int_{0}^{𝑟} \frac{𝑛 (𝑡)}{𝑡} d 𝑡 \leq log 𝑀 (2 𝑟) .

UC Irvine Fall 2025, 4 Complex analysis, entire functions, extremal problems, self-maps

Prove that for every entire function $𝑓 (𝑧)$ ,

max_{| 𝑧 | = 1} | 𝑧 - 𝑓 (\frac{1}{𝑧}) | \geq 1 .

UC Irvine Fall 2025, 5 Complex analysis, meromorphic functions, residues, Mittag-Leffler

Let $𝑎 \in ℂ$ with $| 𝑎 | > 1$ . Determine all meromorphic functions $𝑓$ on $ℂ$ whose only singular points are simple poles at $𝑎^{𝑛}$ for $𝑛 = 1, 2, 3, \dots$ , with

{Res}_{𝑧 = 𝑎^{𝑛}} 𝑓 (𝑧) = 𝑎^{- 𝑛} .

Repeat the problem with the residue condition replaced by

{Res}_{𝑧 = 𝑎^{𝑛}} 𝑓 (𝑧) = 𝑎^{𝑛} .

UC Irvine Fall 2025, 6 Complex analysis, uniform convergence, normal convergence, entire functions

Prove that the sequence of entire functions

𝑓_{𝑛} (𝑧) = 𝑧 𝑒^{- 𝑛 𝑧^{2}}

converges uniformly on $ℝ$ , but does not converge normally, that is, uniformly on compact subsets, in the disk $| 𝑧 | < 𝑟$ for any $𝑟 > 0$ .

UC Irvine Fall 2025, 7 Complex analysis, Phragmén–Lindelöf, growth bounds, entire functions

Suppose $𝑓$ is entire and there exists $𝐴 > 0$ such that

| 𝑓 (𝑧) | \leq 𝐴 {(1 + | 𝑧 |)}^{10} 𝑒^{10 Im 𝑧}

for all $𝑧 \in ℂ$ . Show that

𝑓 (𝑧) = 𝑃 (𝑧) 𝑒^{- 𝑖 10 𝑧}

for some polynomial $𝑃 (𝑧)$ of degree at most $10$ .

UC Irvine Fall 2025, 8 Complex analysis, Mittag-Leffler expansion, cotangent, partial fractions

Show that for all $𝑧 \in ℂ \ ℤ$ ,

𝜋 cot (𝜋 𝑧) = \frac{1}{𝑧} + \sum_{𝑛 = 1}^{\infty} \frac{2 𝑧}{𝑧^{2} - 𝑛^{2}} .

Discuss the convergence of the series.

UC Irvine Fall 2025, 9 Complex analysis, Cauchy estimates, Schwarz lemma, growth bounds

Let $𝑓$ be holomorphic on the disk $| 𝑧 | < 2$ . Suppose there exists $𝑁 > 1$ such that for all $𝑟 \in [1, 2)$ ,

max_{| 𝑧 | = 𝑟} | 𝑓 (𝑧) | \leq 𝑟^{𝑁} .

Show that for some universal constant $𝐶 > 0$ , independent of $𝑓$ and $𝑁$ ,

max_{| 𝑧 | = 1} | 𝑓^{'} (𝑧) | \leq 𝐶 𝑁^{2} .

UC Irvine Spring 2024, 1 Complex analysis, harmonic functions, zero sets, isolated points

Let $𝑢$ be a real-valued harmonic function on a domain $𝑈 \subset ℂ$ . Show that the zero set of $𝑢$ does not have an isolated point. The zero set is

𝑍 = {𝑧 \in 𝑈 : 𝑢 (𝑧) = 0} .

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